GRADING AND SCREENING 233 use of screens. (3) Satisfactory sizing of oblong, spindle shaped or flat particles such as oats the close sizing of which is impossible on screens. (4) The elimination of the influence of specific gravity. It is possible to grade a mixture of shot of various size and cork. (5) Comminution of friable material is usually less than with screening devices. SCREENS AND SCREENING Theory of Screening with Flat Screens.—The splitting of loose fragmental or granular material into three or more sizes and according to one or two linear dimensions of the screen aperture is almost universally done by screens, and these devices are by far the most important ones in the whole field of grading. If the components of a fragmental or granular mass are isometric such as are cubes, spheres, octahedrons, etc., the characteristic of a perfect or theoretical grading resulting from treatment in a battery of screens is the presence of individual particles whose axial1 measure is but little less than the width of aperture of the two screens which, take part in the grading. If the aperture be square, the axial measures of the limiting grains will be a trifle less than the linear dimensions of the sides of the two squares. If the aperture be oblong, the limiting criterion will be the width of the oblong openings. If the holes in the screens be round, the largest and smallest grains in the perfect grading will have axial measures but little less than 0.707 times the diameter of large and small openings respectively. Where the grains are spherical the limits will be but little less than the diameters. If the grains be prisms, ellipsoids, etc., with two axes of the same length but the third longer, the criterion of the limits of the grains appearing in a particular grading will be the dimension of the short axes in the case of square and round holes, or all three of the axes if the apertures in the screen be slotted. Further descriptions of the limits of grading for regular figures might be stated, but these will suffice, for with very little extension of thought they will cover all regular figures. If a grading be of fragmental material such as would result from crushing rock, then it is evident that if one axis is selected in a particle of such a grading, any other pair at right angles to one another and the first will within certain limits, vary as to length, the length depending upon the point they intersect the first axis. Also as the first axis is rotated in various planes its length will change within certain limits as well as the two axes at right angles to it. The limits of the length of the axes are of course the boundary of the particle wherever they happen to pierce it. What then is the criterion for the upper and lower limits of grading for material of this kind ? It is impossible to state it in precise language; but quite evidently, fixing the mind on one axis as the direction axis in which a fragment is passing through an aperture of the screen controlling the upper limit of a grading, then all the other pairs at right angles to it and for any point of intersection with the first must define all planes which will bo bounded by the aperture figure. Otherwise it is evident the grain will be unable to pass through. The axes of course do not, except accidentally, correspond with the extreme length of the planes which they define owing to the irregularity of the outline of the grains. The extreme upper limit will be the dimensions of a fragment measured at right angles to the only direction axis though not to each other, except accidentally, along which the fragment can just pass through the larger aperture of the two engaged in a grading. Evidently somewhere along such a direction axis there is a plane at rig-lit angles to it which almost makes contact at three or more points with the sides of the aperture. By a similar course of reasoning the lower limit of grading for a fragmental grading can be made clear or understandable. Nothing has been said in this discussion on the screening limits of fragmental grains as to their shape. They must have sensible length, 1 The axes are taken in the same position as mineralogical axes.